Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously, generalizing proper interval graphs, and a larger family of NP-complete problems are known to be polynomially solvable for graphs with bounded proper thinness. It is known that the thinness of a graph is at most its pathwidth plus one. In this work, we prove that the proper thinness of a graph is at most its bandwidth, for graphs with at least one edge. It is also known that boxicity is a lower bound for the thinness. The main results of this work are characterizations of 2-thin and 2-proper thin graphs as intersection graphs of rectangles in the plane with sides parallel to the Cartesian axes and other specific conditions. We also bound the bend number of graphs with low thinness as vertex intersection graphs of paths on a grid ($B_k$-VPG graphs are the graphs that have a representation in which each path has at most $k$ bends). We show that 2-thin graphs are a subclass of $B_1$-VPG graphs and, moreover, of monotone L-graphs, and that 3-thin graphs are a subclass of $B_3$-VPG graphs. We also show that $B_0$-VPG graphs may have arbitrarily large thinness, and that not every 4-thin graph is a VPG graph. Finally, we characterize 2-thin graphs by a set of forbidden patterns for a vertex order.